Computational algebraic number theory has been attracting broad interest in the last few years due to its potential applications in coding theory and cryptography. For this reason, the Deutsche Mathematiker Vereinigung initiated an introductory graduate seminar on this topic in Düsseldorf. The lectures given there by the author served as the basis for this book which allows fast access to the state of the art in this area. Special emphasis has been placed on practical algorithms - all developed in the last five years - for the computation of integral bases, the unit group and the class group of arbitrary algebraic number fields. Contents: Introduction

Topics from finite fields

Arithmetic and polynomials

Factorization of polynomials

Topics from the geometry of numbers

Hermite normal form

Lattices

Reduction

Enumeration of lattice points

Algebraic number fields

Introduction

Basic Arithmetic

Computation of an integral basis

Integral closure

Round-Two-Method

Round-Four-Method

Computation of the unit group

Dirichlet's unit theorem and a regulator bound

Two methods for computing r independent units

Fundamental unit computation

Computation of the class group

Ideals and class number

A method for computing the class group

Appendix

The number field sieve

KANT

References

Index

...read more